A Mathematical Model of Spatial Self-Organization in a Mechanically Active Cellular Medium

A general continual model of a medium composed of mechanically active cells is proposed. The medium is considered to be formed by three phases: cells, extracellular fluid, and an additional phase that is responsible for active interaction forces between cells and, for instance, may correspond to a system of protrusions that provide the development of active contractile forces. The deformation of the medium, which is identified with the deformation of the cell phase, consists of two components: elastic deformation of individual cells and cell rearrangements. The elastic deformation is associated with stresses in the cell phase. The spherical component of the stress tensor describes the nonlinear resistance of the cellular medium, which leads to the impossibility of its excessive compression. The constitutive equation for pressure in the cell phase is taken in the form of a nonlinear dependence on the volume cell density. The rearrangement of cells is considered as a flow controlled by stresses in the cell phase, active stresses, and fluid pressure. The tensor of active stresses is assumed to be spherical and nonlocally dependent on the cell density. Assuming that the process of biological tissue deformation is slow, we obtained a reduced model that neglects the elastic deformation of cells, compared to the inelastic deformation. A linear stability analysis of a spatially uniform steady-state solution was performed. The hydrostatic pressure of fluid is present among the parameters that are responsible for the loss of stability of the steady-state solution: an increase in it has a destabilizing effect owing to the action of the component of the interphase interaction force that is determined by the fluid pressure. The model we obtained can be used to describe the process of cavity formation in an initially homogeneous cell spheroid. The role of local and nonlocal mechanisms of active stress generation in the formation of cavity is investigated.     

Hydrophobic interaction and a model for the elasticity of elastin

The thermodynamics of the elastic process in the rubberlike protein elastin have been investigated by microcalorimetery. The results indicate that the reversible heat liberated upon the extension of water-swollen elastin at room temperature is much largerthan the stored elastic energy, indicating a large than the stored elastic energy, indicating a large, negative internal energy change for stretching. The ratio of the measured internal energy change to the stored energy varies inversely wiht extension, and at 22° C it is ?91 for 2% extension and ?3 for 70% extension. The interanl energy change also varies dramatically with temperature over the range of 2–65° C it is zero. The temperature dependence for internal energy change is virtually identical to the temperature dependence for internal energy changes associated with the breaking of hydrophobic interactions, and it is suggested that the measured internal energy change can be attributed entirely to hte absorption of water onto nonpolar groups in the elastin network. Calculatons based on this assumption indicate that the free-energy change associated with this solvent–polymer process is large and positive. It is concluded that the absorption of water onto hydrophobic groups contributes to the elasticity of elastin, particularly at extensions of less than about 70%. The implications of this elastic mechanism are discussed in terms of the random-network model for elastin structure.     

Solvent reorganization contribution to the transfer thermodynamics of small nonpolar molecules.

The experimental thermodynamic data for the dissolution of five simple hydrocarbon molecules in water were combined with the solute-solvent interaction energy from a computer simulation study to yield data on the enthalpy change of solvent reorganization. Similar data were generated for dissolving these same solute molecules in their respective neat solvents using the equilibrium vapor pressure and the heat of vaporization data for the pure liquid. The enthalpy and the free energy changes upon cavity formation were also estimated using the temperature dependence of the solute-solvent interaction energy. Both the enthalpy and T delta S for cavity formation rapidly increase with temperature in both solvent types, and the free energy of cavity formation can be reproduced accurately by the scaled particle theory over the entire temperature range in all cases. These results indicate that the characteristic structure formation around an inert solute molecule in water produces compensating changes in enthalpy and entropy, and that the hydrophobicity arises mainly from the difference in the excluded volume effect.     

A desolvation barrier to hydrophobic cluster formation may contribute to the rate-limiting step in protein folding.

To gain insight into the free energy changes accompanying protein hydrophobic core formation, we have used computer simulations to study the formation of small clusters of nonpolar solutes in water. A barrier to association is observed at the largest solute separation that does not allow substantial solvent penetration. The barrier reflects an effective increase in the size of the cavity occupied by the expanded but water-excluding cluster relative to both the close-packed cluster and the fully solvated separated solutes; a similar effect may contribute to the barrier to protein folding/unfolding. Importantly for the simulation of protein folding without explicit solvent, we find that the interactions between nonpolar solutes of varying size and number can be approximated by a linear function of the molecular surface, but not the solvent-accessible surface of the solutes. Comparison of the free energy of cluster formation to that of dimer formation suggests that the assumption of pair additivity implicit in current protein database derived potentials may be in error.     

Erythropoietin unfolding: thermodynamics and its correlation with structural features

Human erythropoietin (EPO) is a glycoprotein hormone considered to be the principal regulator of red blood cell formation. Although its recombinant version (rEPO) has been widely used for treatment of various anemias and its biological effects are relatively well-known, we know little about its biophysical properties and their relation to its structure. To gain a fuller understanding of the structural and functional properties of rEPO on the molecular level we followed its thermal and urea-induced unfolding at different pH (3.1-9.4) and urea concentrations (0-8 M) using spectropolarimetry, UV absorption, intrinsic emission fluorescence, and differential scanning calorimetry. Our results show that under a variety of conditions rEPO undergoes thermal or urea-induced denaturation that may be considered as a reversible two-state process characterized by unusually high (thermal) or moderate (urea-induced) extent of the residual structure. The highest thermal stability of the protein observed in aqueous solutions at physiological pH appears to be due to the largest difference in the extent of structure in the denatured and native state at this pH. The comparison between experimentally determined energetics of rEPO denaturation and its structure-based calculations indicates that the parametrization of thermodynamic quantities in terms of changes in solvent accessible nonpolar and polar surface areas resulting from protein unfolding can be successfully used provided that these changes are estimated from combination of experimentally determined deltaC(o)p and deltaH(o) values and not calculated from the structure of the protein's folded and assumingly fully unfolded state.     

Heat capacity changes upon burial of polar and nonpolar groups in proteins

In this paper we address the question of whether the burial of polar and nonpolar groups in the protein locale is indeed accompanied by the heat capacity changes, DeltaC(p), that have an opposite sign, negative for nonpolar groups and positive for polar groups. To accomplish this, we introduced amino acid substitutions at four fully buried positions of the ubiquitin molecule (Val5, Val17, Leu67, and Gln41). We substituted Val at positions 5 and 17 and Leu at position 67 with a polar residue, Asn. As a control, Ala was introduced at the same three positions. We also replaced the buried polar Gln41 with Val and Leu, nonpolar residues that have similar size and shape as Gln. As a control, Asn was introduced at Gln41 as well. The effects of these amino acid substitutions on the stability, and in particular, on the heat capacity change upon unfolding were measured using differential scanning calorimetry. The effect of the amino acid substitutions on the structure was also evaluated by comparing the (1)H-(15)N HSQC spectra of the ubiquitin variants. It was found that the Ala substitutions did not have a considerable effect on the heat capacity change upon unfolding. However, the substitutions of aliphatic side chains (Val or Leu) with a polar residue (Asn) lead to a significant (> 30%) decrease in the heat capacity change upon unfolding. The decrease in heat capacity changes does not appear to be the result of significant structural perturbations as seen from the HSQC spectra of the variants. The substitution of a buried polar residue (Gln41) to a nonpolar residue (Leu or Val) leads to a significant (> 25%) increase in heat capacity change upon unfolding. These results indicate that indeed the heat capacity change of burial of polar and nonpolar groups has an opposite sign. However, the observed changes in DeltaC(p) are several times larger than those predicted, based on the changes in water accessible surface area upon substitution.     

Energetics of complementary side-chain packing in a protein hydrophobic core

The energetics of complementary packing of nonpolar side chains in the hydrophobic core of a protein were analyzed by protein engineering experiments. We have made the mutations Ile----Val, Ile----Ala, and Leu----Ala in a region of the small bacterial ribonuclease barnase where the major alpha-helix packs onto the central beta-sheet. The destabilization resulting from the creation of cavities was determined by measuring the decrease in free energy of folding from reversible denaturation induced by urea, guanidinium chloride, or heat. The different methods give consistent and reproducible results. The loss in free energy of folding for the mutant proteins is 1.0-1.6 kcal/mol per methylene group removed. This exceeds by severalfold the values obtained from model experiments of the partitioning of relevant side chains between aqueous and nonpolar solvents. Much of this discrepancy arises because two surfaces are buried when a protein folds--both the amino acid side chain in question and the portions of the protein into which it packs. These experiments directly demonstrate that the interior packing of a protein is crucial in stabilizing its three-dimensional structure: the conversion of leucine or isoleucine to alanine in the hydrophobic core loses half the net free energy of folding of barnase with a concomitant decrease in yield of the expressed recombinant protein.     

Thermodynamic stability of the C-terminal domain of the human inducible heat shock protein 70

The stability of the substrate-binding region of human inducible Hsp70 was studied by a combination of spectroscopic and calorimetric methods. Thermal denaturation of the protein involves four accessible states: the native state, two largely populated intermediates, and the denatured state, with transition temperatures of 52.8, 56.2 and 71.2 degrees C, respectively, at pH 6.5. The intermediate spectroscopic properties resemble those of molten globules but they still retain substantial enthalpy and heat capacity of unfolding. Moreover, the similar heat capacities of the first intermediate and the native state suggests that the hydrophobic core of the intermediate would be highly native-like and that its formation would involve an increased disorder in localized portions of the structure rather than formation of a globally disordered state. The structure of the C-terminal of Hsp70 is destabilized as the pH separates from neutrality. The intermediates become populated under heat shock conditions at acidic and basic pHs. Denaturation by guanidine chloride also indicated that the protein undergoes a sequential unfolding process. The free energy change associated to the loss of secondary structure at 20 degrees C (pH 6.5) is 3.1 kcal.mol(-1) at high salt conditions. These values agree with the free energy changes estimated from differential scanning calorimetry for the transition between the second intermediate and the final denatured state.     

Molecular Dynamics Free Energy Calculations to Assess the Possibility of Water Existence in Protein Nonpolar Cavities

Are protein nonpolar cavities filled with water molecules? Although many experimental and theoretical investigations have been done, particularly for the nonpolar cavity of IL-1β, the results are still conflicting. To study this problem from the thermodynamic point of view, we calculated hydration free energies of four protein nonpolar cavities by means of the molecular dynamics thermodynamic integration method. In addition to the IL-1β cavity (69 ų), we selected the three largest nonpolar cavities of AvrPphB (81 ų), Trp repressor (87 ų), and hemoglobin (108 ų) from the structural database, in view of the simulation result from another study that showed larger nonpolar cavities are more likely to be hydrated. The calculations were performed with flexible and rigid protein models. The calculated free energy changes were all positive; hydration of the nonpolar cavities was energetically unfavorable for all four cases. Because hydration of smaller cavities should happen more rarely, we conclude that existing protein nonpolar cavities are not likely to be hydrated. Although a possibility remains for much larger nonpolar cavities, such cases are not found experimentally. We present a hypothesis to explain this: hydrated nonpolar cavities are quite unstable and the conformation could not be maintained.     

Flow-induced deformation from pressurized cavities in absorbing porous tissues

The behaviour of a cavity during an injection of fluid into biological tissue is considered. High cavity pressure drives fluid into the neighbouring tissue where it is absorbed by capillaries and lymphatics. The tissue is modelled as a nonlinear deformable porous medium with the injected fluid absorbed by the tissue at a rate proportional to the local pressure. A model with a spherical cavity in an infinite medium is used to find the pressure and displacement of the tissue as a function of time and radial distance. Analytical and numerical solutions for a step change in cavity pressure show that the flow induces a radial compression in the medium together with an annular expansion, the net result being an overall expansion of the medium. Thus any flow induced deformation of the material will aid in the absorption of fluid.     

Non-Newtonian flow-induced deformation from pressurized cavities in absorbing porous tissues

We investigate the behavior of a spherical cavity in a soft biological tissue modeled as a deformable porous material during an injection of non-Newtonian fluid that follows a power law model. Fluid flows into the neighboring tissue due to high cavity pressure where it is absorbed by capillaries and lymphatics at a rate proportional to the local pressure. Power law fluid pressure and displacement of a solid in the tissue are computed as function of radial distance and time. Numerical solutions indicate that shear thickening fluids exhibit less fluid pressure and induce small solid deformation as compared to shear thinning fluids. Absorption in the biological tissue increases as a consequence of flow-induced deformation for power law fluids. In most cases non-Newtonian results are compared with the viscous fluid case to magnify the differences.     

Determination of contact angle by molecular simulation using number and atomic density contours

The contact angles of Lennard-Jones fluid droplets on a structureless solid surface, simulated using Monte Carlo simulation, are calculated by fitting isochoric surfaces and making a number of assumptions about the droplet. The results show that there are significant uncertainties in the calculated contact angles due to the choice of these assumptions, such as the grid size used in tracking the isochoric density profile, the omission of isochoric data points near the surface and the function used to fit the isochoric profile. In this study, we propose a new method of calculating density contours based on atomic density instead of number density. This method results in a much smaller variation in contact angle when applying different assumptions than using number density for isochoric contours. The most consistent results, across a range of assumptions about the droplet and the contact angle, come from averaging the contact angle from several isochoric density profiles. In addition, this gives a measurement of the variation due to the choice of isochoric density.     

Ligand binding to the voltage-gated Kv1.5 potassium channel in the open state--docking and computer simulations of a homology model

The binding of blockers to the human voltage-gated Kv1.5 potassium ion channel is investigated using a three-step procedure consisting of homology modeling, automated docking, and binding free energy calculations from molecular dynamics simulations, in combination with the linear interaction energy method. A reliable homology model of Kv1.5 is constructed using the recently published crystal structure of the Kv1.2 channel as a template. This model is expected to be significantly more accurate than earlier ones based on less similar templates. Using the three-dimensional homology model, a series of blockers with known affinities are docked into the cavity of the ion channel and their free energies of binding are calculated. The predicted binding free energies are in very good agreement with experimental data and the binding is predicted to be mainly achieved through nonpolar interactions, whereas the relatively small differences in the polar contribution determine the specificity. Apart from confirming the importance of residues V505, I508, V512, and V516 for ligand binding in the cavity, the results also show that A509 and P513 contribute significantly to the nonpolar binding interactions. Furthermore, we find that pharmacophore models based only on optimized free ligand conformations may not necessarily capture the geometric features of ligands bound to the channel cavity. The calculations herein give a detailed structural and energetic picture of blocker binding to Kv1.5 and this model should thus be useful for further ligand design efforts.     

Role of hydration water in protein unfolding

In this paper, following our work on the two-state outer neighbor mixed bonding model of water, it is proposed that polar groups promote the formation of the low density ice Ih-type bonding in their neighborhood, whereas nonpolar groups tend to promote the higher density ice II-type structure. In a protein, because of the large numbers of exposed polar and nonpolar groups, large changes in the neighboring water structure can occur. These changes, of course, depend on whether the protein is in its native or its unfolded state and will be shown here to have a direct impact on the thermodynamics of protein unfolding at both high and low temperatures. For example, it is known that the polar hydration entropies become rapidly more negative with increasing temperature. This very unusual behavior can be directly related to the promotion in the outer bulk liquid of the more stable Ih-type bonding at the expense of II-type bonding by polar groups of the protein. In contrast, nonpolar groups have an opposite effect on the thermodynamics. It is the delicate balance created by these outer hydration contributions, mixed with ordinary thermodynamic contributions from the inner hydration shell and those from hydrogen-bond and van der Waals forces within the protein molecule itself that is responsible for both heat and cold denaturation of proteins.     

An elongation model of left ventricle deformation in diastole

A numerical method of the left ventricle (LV) deformation, an elongation model, was put forth for the study of LV fluid mechanics in diastole. The LV elongated only along the apical axis, and the motion was controlled by the intraventricular flow rate. Two other LV models, a fixed control volume model and a dilation model, were also used for model comparison and the study of LV fluid mechanics. For clinical sphere indices (SIs, between 1.0 and 2.0), the three models showed little difference in pressure and velocity distributions along the apical axis at E-peak. The energy dissipation was lower at a larger SI in that the jet and vortex development was less limited by the LV cavity in the apical direction. LV deformation of apical elongation may represent the primary feature of LV deformation in comparison with the secondary radial expansion. The elongation model of the LV deformation with an appropriate SI is a reasonable, simple method to study LV fluid mechanics in diastole.     

Reconstitution of clathrin-coated bud and vesicle formation with minimal components

During the process of clathrin-mediated endocytosis an essentially planar area of membrane has to undergo a gross deformation to form a spherical bud. Three ways have been recognized by which membranes can be induced to transform themselves locally from a planar state to one of high curvature: a change in lipid distribution between the leaflets, insertion of a protein into one leaflet and formation of a protein scaffold over the surface. Such a scaffold is spontaneously generated by clathrin. Conjectures that the attachment of clathrin was the cause of the change in curvature were challenged on theoretical grounds, and also by the discovery of a number of clathrin-associated proteins with the capacity to induce membrane curvature. We have now developed a cell-free system that has enabled us to demonstrate that clathrin polymerization alone is sufficient to generate spherical buds in a membrane. This process is reversible, as shown by the reassimilation of the buds into the planar membrane when the intra-clathrin contacts are dissociated by the chaperone Hsc70. We further show that the final step in the formation of coated vesicles ensues when clathrin-coated buds are released through the action of dynamin.     

Thermodynamics of the reconstitution of tuna cytochrome c from two peptide fragments

Two peptide fragments from tuna cytochrome c (cyt c), N-fragment (residues 1-44 containing the heme) and C-fragment (residues 45-103), combine to form a 1:1 fragment complex. This was clearly proved by ion-spray mass spectrometry. It was found from CD and NMR spectra that the structure of the fragment complex formed is similar to that of an intact cyt c, although each isolated fragment itself is unstructured. Binding constants and enthalpies upon the complex formation were directly observed by isothermal titration calorimetry. Thermodynamic parameters (deltaG(o)b, deltaHb, deltaS(o)b, and deltaC(b)p)) associated with the complex formation were determined at various pHs and temperatures. DeltaHb was found to be almost independent of pH values. The change in heat capacity accompanying the complex formation (deltaC(b)p) was directly determined from the temperature dependence of deltaHb. In addition, the change in heat capacity and enthalpy upon tuna cyt c unfolding were determined by differential scanning calorimetry. Thermodynamic parameters for the unfolding/dissociation process of the fragment complex were compared with those for cyt c unfolding at pH 3.9 and 303 K. In a comparison of two unfolding processes, the heat capacity change of each was very close to the other, while both the unfolding enthalpy and entropy of the fragment complex were larger than those of tuna cyt c. These thermodynamic data suggest that the internal interactions between polar groups (hydrogen bonding) and nonpolar groups (van der Waals interactions) are preserved in the fragment complex as well as in the native state of cyt c.     

Protein compressibility, dynamics, and pressure

The relationship between the elastic and dynamic properties of native globular proteins is considered on the basis of a wide set of reported experimental data. The formation of a small cavity, capable of accommodating water, in the protein interior is associated with the elastic deformation, whose contribution to the free energy considerably exceeds the heat motion energy. Mechanically, the protein molecule is a highly nonlinear system. This means that its compressibility sharply decreases upon compression. The mechanical nonlinearity results in the following consequences related to the intramolecular dynamics of proteins: 1) The sign of the electrostriction effect in the protein matrix is opposite that observed in liquids-this is an additional indication that protein behaves like a solid particle. 2) The diffusion of an ion from the solvent to the interior of a protein should depend on pressure nonmonotonically: at low pressure diffusion is suppressed, while at high pressure it is enhanced. Such behavior is expected to display itself in any dynamic process depending on ion diffusion. Qualitative and quantitative expectations ensuing from the mechanical properties are concordant with the available experimental data on hydrogen exchange in native proteins at ambient and high pressure.     

Binding Thermodynamics of the Transition State Analogue Coformycin and of the Ground State Analogue 1-Deazaadenosine to Bovine Adenosine Deaminase

Binding of the transition state analogue coformycin and the ground state analogue 1-deaazadenosine to bovine adenosine deaminase have been thermody-namically characterized. The heat capacity changes for coformycin and 1-deazaadenosine binding are - 4.7 × 0.8 kJ/mole-K and -1.2 × 0.1 kJ/mole-K, respectively. Since the predominant source of heat capacity change in enzyme interactions are changes in the extent of exposure of nonpolar amino acid side chains to the aqueous environment and the hydrophobic effect is the predominant factor in native structure stabilization, we propose that the binding of either class of ligand is associated with a stabilizing enzyme conformational change with coformycin producing the far greater effect Analysis of the T dependence of the second order rate constant for formation of the enzyme/coformycin complex further reveals that the conformational change is not rate limiting. We propose that the enzyme may facilitate catalysis via the formation of a stabilizing conformation at the reaction transition state.     

Analysis of isochoric subcooling

Because ice-I is less dense than water, the formation of an ice nucleus in an isochoric (constant volume) chamber will cause an increase in pressure. This analysis shows that the energy required to overcome such a pressure increase makes homogeneous ice nucleation thermodynamically improbable in an isochoric system at temperatures above -109 degrees C. By suppressing ice nucleation, isochoric cooling is expected to significantly promote vitrification. Because water has a higher freezing temperature and a lower glass-transition temperature than physiological solutions, this analysis represents a scenario for avoiding ice crystallization during the preservation of biological substances. While isochoric cryopreservation has not yet been put into practice, this theoretical, first-order analysis suggests that if attainable it could make organ preservation significantly more effective and practical.